Theorem. the eigenspace decomposition of a module [fgap-000Z]
AGENTDRAFTED
Let \(z\) be a central involution in \(A\), let
\(e_+,e_-\) be its associated idempotents, and let \(M\) be a left
\(A\)-module. Left multiplication defines \(A\)-linear projections
\[
p_\pm :M\longrightarrow M,\qquad p_\pm (m)=e_\pm m.
\]
They satisfy
\[
p_\pm ^2=p_\pm ,\qquad
p_++p_-=\mathrm {id}_M,\qquad
p_+p_-=p_-p_+=0.
\]
Their ranges and kernels are
\[
\begin {aligned}
\operatorname {range}(p_+)&=e_+M,&
\ker (p_+)&=\operatorname {range}(p_-)=e_-M,\\
\operatorname {range}(p_-)&=e_-M,&
\ker (p_-)&=\operatorname {range}(p_+)=e_+M.
\end {aligned}
\]
Consequently, if
\[
M_+=e_+M,\qquad M_-=e_-M,
\]
then
\[
M=M_+\oplus M_-.
\]
These summands are precisely the two eigenspaces for the action of \(z\):
\[
M_+=\{m\in M:zm=m\},\qquad
M_-=\{m\in M:zm=-m\}.
\]
Idempotents as projections and complementary orthogonal idempotents as
direct decompositions are recorded in
[webb2007finite, exercise 2.7, p. 15]; see also
[sengupta2010representations, props. 4.5.1--4.5.3, pp. 63--67].